Topological insulators represent a unique class of materials characterized by an insulating bulk and conducting surface states protected by time-reversal symmetry. The electronic structure of these surface states is distinguished by the presence of Dirac fermions, which exhibit linear energy-momentum dispersion, forming a Dirac cone similar to graphene. However, unlike graphene, topological surface states possess strong spin-momentum locking, a feature that arises from spin-orbit coupling and underpins their topological protection.

The Dirac cone in topological insulators emerges at the surface due to the inversion of bulk bands driven by spin-orbit interactions. In materials like Bi₂Se₃ and Sb₂Te₃, the surface states form a single Dirac cone at the Γ point in the Brillouin zone. The linear dispersion relation near the Dirac point can be expressed as E(k) = ±ħv_F|k|, where v_F is the Fermi velocity and k is the momentum relative to the Dirac point. This linearity is a hallmark of massless Dirac fermions, analogous to relativistic particles. However, unlike graphene, where the Dirac cones are degenerate in spin, topological surface states exhibit a helical spin texture where the electron spin is locked perpendicular to its momentum. This spin-momentum locking ensures that backscattering is suppressed, as any reversal of momentum would require a reversal of spin, which is forbidden in the absence of magnetic impurities or breaking of time-reversal symmetry.

Graphene’s Dirac cones, located at the K and K' points of its hexagonal Brillouin zone, also host massless Dirac fermions with linear dispersion. However, the spin degeneracy in graphene means that spin and momentum are not intrinsically coupled. While graphene’s charge carriers can exhibit pseudospin-momentum locking due to the sublattice symmetry, this is distinct from the real spin-momentum locking in topological insulators. The absence of strong spin-orbit coupling in graphene means that its Dirac cones lack the topological protection seen in materials like Bi₂Se₃.

Angle-resolved photoemission spectroscopy (ARPES) has been instrumental in verifying the existence of Dirac fermions in topological insulators. In Bi₂Se₃, ARPES measurements reveal a single Dirac cone at the Γ point with a Fermi velocity of approximately 5 × 10⁵ m/s. The spin-resolved ARPES further confirms the spin-momentum locking, showing that the in-plane spin polarization reverses when the momentum direction is inverted. Similarly, in Sb₂Te₃, ARPES data demonstrates a Dirac cone with a slightly higher Fermi velocity of around 6 × 10⁵ m/s, along with the characteristic spin texture. These measurements provide direct evidence of the topological nature of the surface states, distinguishing them from trivial two-dimensional electron gases or graphene’s Dirac cones.

The robustness of topological surface states against non-magnetic perturbations is another key difference from graphene. In graphene, disorder or impurities can lead to localization effects and scattering, whereas topological surface states remain conducting as long as time-reversal symmetry is preserved. This property makes topological insulators promising for applications in spintronics and quantum computing, where dissipationless spin currents are desirable.

The unique electronic structure of topological surface states also has implications for their interaction with external fields. For instance, applying a magnetic field can break time-reversal symmetry, opening a gap in the Dirac cone. In contrast, graphene’s Dirac cones can be gated electrostatically, but their response to magnetic fields does not involve the same topological considerations. Additionally, the presence of higher-order spin-orbit coupling terms in topological insulators can lead to warping effects in the Dirac cone at higher energies, a feature absent in graphene’s more symmetric dispersion.

In summary, while both topological insulators and graphene host Dirac fermions with linear dispersion, the spin-momentum locking and topological protection in the former set them apart. ARPES studies on Bi₂Se₃ and Sb₂Te₃ have provided conclusive evidence of these properties, solidifying our understanding of their electronic structure. The differences between these systems highlight the rich physics arising from strong spin-orbit coupling and have significant implications for future electronic and spintronic devices.

The exploration of topological surface states continues to reveal new phenomena, such as the interplay with superconductivity and magnetism, further expanding their potential applications. As research progresses, the distinct features of these materials will likely lead to breakthroughs in low-power electronics and quantum technologies, leveraging their unique electronic properties.